At the 32min 33 sec mark of the video, you can see Charles using his own pen, which has a screw-cap. He dips it into an inkwell, which is kind of odd because it's a fountain pen.
Right after that, William and then Camilla each use a Pilot V-Pen taken from a glass tray with other pens of the same model. I laughed when Camilla tried to screw the cap of the Pilot back on, not realizing the cap is friction-fit.
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